# Thread: [SOLVED] Population of bugs Growth Rate

1. ## [SOLVED] Population of bugs Growth Rate

Suppose that the number of bugs in a rainforest is growing exponentially. There were 2880 bugs after 3 weeks and 5120 bugs after 5 weeks.
1) Find the population growth rate r.
2)How many bugs were there initially?
3)Write an equation that models the number of bugs, n, with the respect to the number of weeks and use it to determine how long it will take for the population of bugs to grow to 10000.

2. Originally Posted by moonman
Suppose that the number of bugs in a rainforest is growing exponentially. There were 2880 bugs after 3 weeks and 5120 bugs after 5 weeks.
...
3)Write an equation that models the number of bugs, n, with the respect to the number of weeks and use it to determine how long it will take for the population of bugs to grow to 10000.
Let t denote the number of weeks and A the initial value at t = 0.

$\displaystyle n(t)=A \cdot e^{k \cdot t}$

Plug in the given values to determine A and k:

$\displaystyle 2880=A \cdot e^{k \cdot 3}~\implies~A=\dfrac{2880}{e^{3k}}$

$\displaystyle 5120 = A \cdot e^{k \cdot 5} ~\implies~5120 = \dfrac{2880}{e^{3k}} \cdot e^{5k}$

Solve for k. I've got $\displaystyle k = \ln\left(\frac43\right)$

and consequently A= 1215

Now solve the equation for t:

$\displaystyle 10000=1215 \cdot e^{\ln\left(\frac43\right) \cdot t}~\implies~ 10000 = 1215 \cdot \left(\frac43\right)^t$

I've got $\displaystyle t \approx 7.33\ weeks$